Tension-shear Coupling Research Articles

An Anisotropic Hyperelastic Constitutive Model with Tension–Shear Coupling for Woven Composite Reinforcements

An anisotropic hyperelastic constitutive model with tension–shear coupling was developed for woven composite reinforcements based on fiber reinforced continuum mechanics theory. The strain energy of the model was additively decomposed into two parts nominally representing the fiber stretches and fiber–fiber interaction considering shear–tension coupling, respectively. Experimental data were used to identify material parameters orderly and simply in the constitutive model for a specific balanced plain woven carbon fabric. The developed model was validated by comparing numerical results with picture-frame shear tests under different pre-stretch ratios, and was then applied to the simulation of a hemispherical stamping experiment, demonstrating that the developed constitutive model is highly suitable to characterize the nonlinear and anisotropic mechanical behaviors of woven composite reinforcements under large deformation. The proposed model establishes a theoretical foundation for more accurate forming simulation and processing optimization of woven fabric composites.

Understanding different types of coupling in mechanical behavior of woven fabric reinforcements: A critical review and analysis

This study attempts to provide an improved insight into the significant role of inherent coupling in the mechanical behavior of woven fabric composites. In particular, applying yarns tension to postpone the wrinkling defect has now become a common technique in industry during shear forming of fabric reinforcements. Yet, in-depth understanding of this coupling characteristics is rather limited. The article first distinguishes between the meso-level (inherent) coupling and the macro-level coupling expressed by the general Hook’s law. Then, specific types of inherent couplings are identified, including tension–tension, tension-shear, and shear-tension. A fixture capable of performing combined loadings was employed to characterize the coupling modes in a typical glass/polypropylene plain weave. Results revealed a highly dominant influence of tension-shear coupling on the effective mechanical properties of the fabric, followed by the tension-tension and shear-tension couplings. Discussions are made as to how these macro-level observations may be linked to underlying deformation mechanisms at lower material scales.

Analysis of a two-way tension-shear coupling in woven fabrics under combined loading tests: Global to local transformation of non-orthogonal normalized forces and displacements

Dependence of shear rigidity of woven fabrics on yarn pre-tension has been reported in the recent literature, however some conflict regarding the trend of this effect is observed. Sources of this conflict are discussed and resolved in the present article using a new characterization framework and a custom-design combined loading fixture. It is shown that in order to correctly characterize the tension-shear coupling behavior in woven fabrics, instead of using global measured data, local normalized forces and displacements should be driven via a non-orthogonal transformation procedure, while considering kinematic force coupling in the setup. In addition, the effect of fabric shear on the tensile behavior of yarns has been investigated, suggesting that the coupling under question is in fact two-way. In particular, results revealed that applying yarn pre-tension increases the shear resistance of the fabric reinforcement, while the tensile behavior of the material becomes more compliant when undergoing shear deformation.

Interplanar potential for tension-shear coupling at grain boundaries derived from ab initio calculations

Based on ab initio density functional theory (DFT) calculations we derive an analytical expression for the interplanar potential of grain boundaries and single crystals as a function of coupled tensile and shear displacements. This energy function captures even details of the grain boundary behaviour, such as the tension-softening of the shear instability of aluminium grain boundaries, with good accuracy. The good agreement between the analytical model and the DFT calculations is achieved by introducing two new characteristic parameters, namely the position of the generalised unstable stacking fault with respect to the stable stacking fault, and the ratio of stable and unstable generalised stacking fault energies. One of the potentials’ parameters also serves as a criterion to judge if a grain boundary deforms via crack propagation or dislocation nucleation. We suggest this potential function for application in continuum models, where constitutive relationships for grain boundaries need to be derived from a sound physical model.

Yield Criterion and Crack Tip Plastic Zone of Nickel-Based Single Crystal

A modified Hill yield criterion considering tension-shear stress coupling was given and material parameters of nickel-based single crystal in this criterion was determined by the Least Square Method. Crack tip plastic zones of nickel-based single crystal plate were analyzed by using this modified yield criterion and the plastic zones corresponding to mixed mode I, II crack, were derived. The crack tip plastic zones of nickel-based single crystal were compared to that of isotropic material. The influences of tension-shear stress coupling, compound ratio of mixed mode I, II crack and temperature on crack tip plastic zones were discussed. The results obtained in this paper indicate that the increasing rate of crack tip plastic zones with compound ratio in the plane stress condition is greater than that in the plane strain condition for nickel-based single crystal and the temperature has greater influence on the size of crack tip plastic zones.

Atomistic simulation study on key factors dominating dislocation nucleation from a crack tip in two FCC materials: Cu and Al

Dislocation nucleations from crack tips in FCC copper and aluminum are studied using atomistic simulations. It is shown that the critical load for dislocation nucleation predicted by Rice’s model (Rice, 1992) based on the Peierls concept of dislocation can either be under- or over-estimated in reference to the simulation results. Such discrepancies have not been fully resolved by existing improved nucleation models, due to the complicated atomic environments at crack tips. Based on our simulation results, it is proposed that such discrepancies can be reconciled by the competition of two coupling processes at a crack tip: the tension-shear coupling, which facilitates the dislocation nucleation, and the nucleation-debonding coupling, which retards the dislocation nucleation. In addition, the two couplings are applied to explain the paradoxical observation: easy dislocation nucleation at a blunted crack tip. The present work provides a detailed picture to justify future improvements on Rice’s model for dislocation nucleation and to accurately predict intrinsic brittle to ductile transition for crystalline materials.

Ab initiostudy of tension-shear coupling in NiAl

Results of a density functional theory study of the coupling between tension and shear processes are presented. The coupling potential was derived from calculations of generalized stacking fault energies in combination with cleavages preopened by tension. The effect of tension-shear coupling is studied for various slip systems of the intermetallic compound NiAl. We derive an atomistic potential for the tension-shear coupling, which accounts for slips associated with partial dislocations. The derivation goes beyond the formulation of Sun et al. [Mater. Sci. Eng., A 170, 67 (1993)] by including the next term of the Taylor expansion of the tension-shear coupling potential. This potential is able to describe the tension softening of ⟨111⟩(110) partial dislocations reasonably well.

Reconciliation of continuum and atomistic models for the ductile versus brittle response of iron

We present a detailed analysis of the ductile versus brittle response of bcc iron containing a sharp crack and show that a continuum model based on the Peierls–Nabarro model for dislocation formation is consistent with our atomistic results. Specifically, we compare continuum predictions for dislocation emission from a crack tip loaded in mode I under plane strain, tensile conditions with atomistic results for a ( 1 0)[1 1 0] crack emitting full edge dislocations in the ⟨1 1 1⟩ {1 1 2} slip system. The simulations are based on an N-body potential of the Finnis–Sinclair type for iron at 0 K, and the continuum dislocation model incorporates recent improvements that account for tension–shear coupling on a prescribed slip plane, as well as the T-stress, in an anisotropic solid. We show that the critical load for dislocation nucleation is influenced by the T-stress (modulated by the level of external stress applied parallel to the crack plane in a biaxially-loaded plate), which possesses a critical value associated with a change of mechanism between dislocation emission and crack extension. The results are consistent with recent preliminary analyses that address the effect of crack size and the role of the T-stress in the ductile versus brittle response of crystals.

Eigenanalysis and continuum modelling of an asymmetric beam-like repetitive structure

An asymmetric repetitive pin-jointed structure, based upon a 3-D NASA framework, is analysed using a state variable transfer matrix technique. A conventional transfer matrix cannot be constructed due to the singularity of one partition of the stiffness matrix; instead, a cell (rather than cross-sectional) state vector consisting of displacements only is employed, leading to a generalised eigenvalue problem. The asymmetry of the structure leads to tension–torsion and bending–shear couplings, which may be explained in terms of the tension–shear coupling of a single face of the structure. Equivalent continuum beam properties and coupling coefficients are determined, and the effect of (a)symmetry discussed as a trade between, for example, tension-Poisson's ratio contraction for a symmetric structure, against tension–torsion coupling for the asymmetric.

Tension–shear coupling in slip and decohesion of iron crystals

Results from atomistic simulations are used to parameterize constitutive relations involving the Frenkel sinusoidal dependence of shear stress on sliding displacement and a Rose–Ferrante–Smith universal binding form for dependence of tensile stress on opening displacement, for cases where the crystal symmetry dictates that a polarity in slip displacement exists.

Local buckling of composite laminar plates with various delaminated shapes

An investigation is carried out to understand the buckling behaviour of local delamination near the surface of fiber reinforced laminated plates under mechanical and thermal loads. The shape of the delaminated region considered is rectangular and triangular. The displacement expression is composed of items with the effect of tension-shear coupling and the effect of bend-torsion coupling. The critical strains of laminated plates with various shaped local delamination and different stacking patterns are obtained by making use of the energy principle. Analytical predictions for delamination buckling loads are shown to correlate well with experimental results for a number of different delamination shapes.

Dislocation nucleation from a crack tip: A formulation based on anisotropic elasticity

The analysts of dislocation emission from a crack tip within the Peierls framework [Rice (1992) J. Mech. Phys. Solids 40, 239–271; Rice et al. (1992) Topics in Fracture and Fatigue, pp. 1–58; Sun et al. (1993) Mater. Sci. Engng A170, 67–85], heretofore developed for isotropic solids, is generalized to take into account elastic anisotropy. An incipient dislocation core, represented in terms of a critical configuration at the crack tip, is determined numerically (most simply in the shear-only version of the model, but also for a combined tension-shear version that includes tension-shear coupling constrained by atomic modeling). These solutions improve upon approximations based on an effective shear stress intensity. For fcc crystals and intermetallics, the nucleation event analysed is that of a set of partial dislocations emitted sequentially. The anisotropic formulation accounts for corrections as large as 30% in the critical value of the stress intensity factor for atomic decohesion, or cleavage. The anisotropic critical crack extension force for dislocation emission may be greater or less than its isotropic counterpart. For an embedded-atom-method (EAM) model of bcc α-Fe, the anisotropic values can be as large as 2.4 times the isotropic ones in one crack orientation; in another crack orientation, the values are as much as 40% less than the isotropic analogs. For fcc structures (EAM nickel, aluminum and Ni 3Al), the difference is within a ± 10–25% range. For silicon, the isotropic formulation is good, with less than a 14% difference from the anisotropic counterpart. The anisotropic effects are found to increase with a standard ratio of elastic anisotropy, and are important for predicting intrinsic ductile versus brittle response.

Estimates from atomic models of tension-shear coupling in dislocation nucleation from a crack tip

The normal stress distribution across a slip plane has the effect of reducing the critical loading required for dislocation emission from a crack tip. The reduction by normal stresses was found to be very significant for Si, based on properties estimated for it using density functional theory, to be large for Fe as modeled by the embedded atom method (EAM), and to be smaller in Al, Ni and ordered Ni 3Al, estimated using the EAM. The general dependence over a wide range on parameters characterizing the tension-shear coupling was also determined. In the context of a Peierls model for dislocation nucleation at a crack tip (J.R. Rice, J. Mech. Phys. Solids, 40 (1992) 239), our approach was to search for onset of the dislocation nucleation instability based on the numerical solution of the system of non-linear integral equations describing an incipient dislocation. The incipient dislocation consists of a distribution of sliding and opening displacements along a slip plane emanating from the crack tip; these displacements are related to the shear and tensile stresses across the slip plane by constitutive relations based on the atomic models mentioned. Results from the atomic models are used to parametrize constitutive relations involving a Frenkel sinusoidal dependence of shear stress on sliding displacement at any fixed opening displacement, and a Rose-Ferrante-Smith universal binding form of dependence of tensile stress on opening displacement at any fixed shear displacement. These relations then enter the system of integral equations, solved numerically, which describe the elasticity solution for a non-uniform distribution of sliding and opening along the slip plane. The results show that tension-shear coupling will often significantly reduce the loading for dislocation emission from the value estimated on the basis of an unstable stacking energy γ us determined with neglect of such coupling, in a shear-only type analysis. For the EAM models of the metals considered, a simple and approximate method to account for the tension effects is to use a modified quantity γ us ( u ∗) , which is an unstable stacking energy for lattice planes which are constrained to a fixed opening Δ θ ∗ , corresponding to that for vinishing normal stress at the unstable shear equilibrium position. Moreover, it is found that the normal stress effect can be described well in these cases by replacing the unstable stacking energy γ us in the shear-only model by a tension softened γ us( Ψ), which depends on the phase angle Ψ of the combined tension-shear loading along the slip plane according to the stress intensity factors of the elastic singular solution. The same simple procedures for accounting for tension effects on nucleation are less suitable for lattices with strong coupling such as Si.